Complete graph

This article defines a property that can be evaluated to true/false for any undirected graph, and is invariant under graph isomorphism. Note that the term "undirected graph" as used here does not allow for loops or parallel edges, so there can be at most one edge between two distinct vertices, the edge is completely described by the vertices it joins, and there can be no edge from a vertex to itself.
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Definition

An undirected graph is termed a complete graph or clique if, for any two distinct vertices, there is an edge between them.

On any vertex set, there is a unique complete graph. Thus, up to graph isomorphism, there is a unique complete graph with any given number of vertices.